While following the directions on a treasure map, a pirate walks 36.6 m north, then turns and walks 6.3 m east.
A.) What is the magnitude of the single straight-line displacement that the pirate could have taken to reach the treasure? Answer in units of m. *The answer to A.) is 36.8 meters.
B.)At what angle with the north would he have to walk? Answer in units of ◦ (degrees)
X = 6.3 m.
Y = 36.6 m.
A. D^2 = X^2 + Y^2.
D^2 = (6.3)^2 + (36.6)^2 = 1379.25
D = 37.1 m.
B. tanA = Y/X = 36.6/6.3 = 5.80952
A = 80.2o, CCW.
A = 90 - 80.2 = 9.8o, East of North.
To answer both parts of the question, we can use the Pythagorean theorem and trigonometry. Here's how:
A.) To find the magnitude of the single straight-line displacement, we need to calculate the length of the hypotenuse of the right triangle formed by the North and East displacements.
1. Using the Pythagorean theorem:
c^2 = a^2 + b^2
where c is the length of the hypotenuse, a is the North displacement (36.6 m), and b is the East displacement (6.3 m).
2. Plugging in the values:
c^2 = (36.6)^2 + (6.3)^2
c^2 = 1339.56 + 39.69
c^2 = 1379.25
3. Taking the square root of both sides:
c = sqrt(1379.25)
c ≈ 37.10 m
Therefore, the magnitude of the single straight-line displacement that the pirate could have taken to reach the treasure is approximately 37.10 meters.
B.) To find the angle with the North, we can use trigonometry. The angle can be calculated using the tangent function as follows:
1. Using the tangent function:
tan(theta) = opposite/adjacent
where theta is the angle we want to find, and the opposite side is the East displacement (6.3 m) while the adjacent side is the North displacement (36.6 m).
2. Plugging in the values:
tan(theta) = 6.3/36.6
3. Solving for theta:
theta ≈ tan^(-1)(6.3/36.6)
theta ≈ 10.13°
Therefore, the angle with the North that the pirate would have to walk is approximately 10.13 degrees.
To find the angle with the north, we can use trigonometry. The angle with the north can be found using the tangent function, which is the ratio of the opposite side to the adjacent side.
Let's call the angle with the north "θ" and the opposite side "y" (which is the eastward displacement of 6.3 m) and the adjacent side "x" (which is the northward displacement of 36.6 m).
We can use the formula:
tan(θ) = y / x
Substituting the given values:
tan(θ) = 6.3 / 36.6
Now, we can take the arctan of both sides to solve for θ:
θ = arctan(6.3 / 36.6)
Using a calculator, we find:
θ ≈ 9.78 degrees
Therefore, the pirate would have to walk at an angle of approximately 9.78 degrees with the north.